A small rolls of the top of a stairway horizontally with a velocity of `4.5 ms^(-1)`. Each step is 0.2 m high and 0.3 m wide. If g is `10ms^(-2)`, then the ball will strike the nth step where `n` is equal to (assume ball strike at the edge of the step).

Options

A. 10
B. 9
C. 8
D. 11

Explanation

To solve the problem, we need to determine the number of steps (n) that the ball will strike when it rolls off the top of a stairway horizontally with a given velocity. The steps have specific dimensions, and we will use kinematic equations to find the answer.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Horizontal velocity of the ball, \( v = 4.5 \, \text{m/s} \)
- Height of each step, \( h = 0.2 \, \text{m} \)
- Width of each step, \( w = 0.3 \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

2. **Determine the Time of Flight (t):**
The ball is projected horizontally, so we need to find the time it takes to fall a vertical distance equal to the height of n steps. The vertical distance (s) after falling for time \( t \) is given by:
\[ s = \frac{1}{2} g t^2 \]
For n steps, the total height fallen is:
\[ s = n \cdot h = n \cdot 0.2 \]
Setting the two equations equal gives:
\[ n \cdot 0.2 = \frac{1}{2} \cdot 10 \cdot t^2 \]
Simplifying this:
\[ n \cdot 0.2 = 5 t^2 \]
Rearranging gives:
\[ t^2 = \frac{0.2n}{5} = \frac{0.04n}{1} \]

3. **Determine the Horizontal Distance (R):**
The horizontal distance traveled by the ball in time \( t \) is given by:
\[ R = v \cdot t \]
The horizontal distance for n steps is:
\[ R = n \cdot w = n \cdot 0.3 \]
Setting the two equations equal gives:
\[ n \cdot 0.3 = 4.5 \cdot t \]

4. **Substituting for t:**
From the previous step, we have \( t = \frac{n \cdot 0.3}{4.5} \). Now, we can substitute this expression for \( t \) back into the equation for \( t^2 \):
\[ t^2 = \left(\frac{n \cdot 0.3}{4.5}\right)^2 \]
Substituting into the equation \( t^2 = 0.04n \):
\[ \left(\frac{n \cdot 0.3}{4.5}\right)^2 = 0.04n \]

5. **Solving for n:**
Squaring the left side:
\[ \frac{n^2 \cdot 0.09}{20.25} = 0.04n \]
Rearranging gives:
\[ n^2 \cdot 0.09 = 0.04n \cdot 20.25 \]
Simplifying:
\[ n^2 \cdot 0.09 = 0.81n \]
Dividing both sides by n (assuming \( n \neq 0 \)):
\[ n \cdot 0.09 = 0.81 \]
Thus:
\[ n = \frac{0.81}{0.09} = 9 \]

### Conclusion:
The ball will strike the 9th step.